Problems

In Fig. 1, \(A\) and \(B\) are fixed points at the same level 6 in. apart, to which are hinged the stiff rods \(AD\), 5 in. long, and \(BF\), 3 in. long. \(C\) is the middle point of \(AD\) and the hinged rods \(DE\), \(EF\) and \(CF\) are each \(2\frac{1}{2}\) in. long. A weight of 10 lb. is hung from the point \(F\) and the system is maintained in equilibrium with \(BF\) horizontal by a vertical force \(P\) acting at \(E\). Determine the magnitude of \(P\) and the force in the rod \(FB\), neglecting the weights of the rods and any friction at the hinges.

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A four-wheeled trolley of weight \(w\) has wheels of radius \(r\) which can turn freely on their axles. The distance between the axles is \(2l\) and the centre of gravity is equidistant from them. The trolley is standing on level ground with the front wheels in contact with a vertical step of height \(h\) (\(< r\)). A horizontal force \(P\) is applied to the trolley at a height \(x\) above the ground, in a direction at right angles to the step, and increased until motion occurs. Show that the front or rear wheels will leave the ground first according as \(x\) is less or greater than $$r + \frac{l(r-h)}{\sqrt{2rh-h^2}}.$$ Determine the value of \(P\) which will just cause motion if \(x = r\).

A uniform circular disc of radius \(r\) has a particle, of mass \(m\), attached to it at a distance \(a\) from its axis. It is caused to roll without slipping in a vertical plane on a rough horizontal surface at a constant angular velocity \(\omega\) by means of a varying horizontal force applied through its axis. Obtain an expression for this force at the instant when the particle is at the above the horizontal.

A ship is observed from a lighthouse in a direction \(30^\circ\) east of north, and at the instant of observation this angle is found to be increasing at the rate of \(6^\circ\) a minute. Ten minutes later the ship is due east of the lighthouse. Calculate the course of the ship, assuming it to be travelling in a straight line at a uniform speed.

Prove that, if \(G\) is the centre of gravity of a plane lamina of mass \(M\) and \(I_G\) is the moment of inertia of the lamina about the line through \(G\) perpendicular to its plane, then its moment of inertia about the parallel line through a point \(P\) in its plane is \(I_G + M \cdot GP^2\). The cross-section of a thin straight open-ended tube is a regular hexagon of side \(a\). It is laid with one face in contact with a rough horizontal plane and the plane is then tilted slowly. Find the angle at which the cylinder at the instant when the adjacent face strikes the plane.

A thin uniform rod \(ABC\) is bent at right angles at \(B\) forming two straight portions \(AB\) and \(BC\), each of length \(l\). It is placed in a vertical plane over two small smooth pegs at the same level and distant \(4\sqrt{2}\) apart, the point \(B\) being uppermost. Show that in the symmetrical position, for displacement in its plane, $$10ga < 16V^2 < 16ga$$ and $$25V = 16V.$$ Show also that if \(l\) is between \(4\sqrt{2}a\) and \(4a\) there will be three positions of equilibrium, two of which will be stable.

A plane lamina is moving in its own plane. Show that in general its motion at any instant can be represented as a rotation about a point (the instantaneous centre of rotation). A thin rod \(PQ\) of mass \(M\) and length \(l\) is constrained to move so that \(P\) and \(Q\) lie on two lines \(OA\) and \(OB\) respectively, where \(\angle AOB = 60^\circ\). At a certain instant the end \(P\) is moving with a velocity \(v\) and \(OP = OQ = l\). Calculate the kinetic energy of the rod.

A trolley, of mass 10 lb., can move freely on a horizontal track. It has a horizontal platform on which rests a particle of mass 10 lb., the coefficient of friction between the particle and the platform being \(\frac{1}{2}\). The system being initially at rest, the trolley is set in motion by a horizontal force which increases from zero to 12 lb.-wt. in 2 sec. at a uniform rate. Draw a graph showing the acceleration of the trolley during this period and determine the velocity of the particle relative to the trolley at the end of the period. [Take \(g\) as 32 ft. per sec. per sec.]

A uniform circular cylinder (Fig. 2) is placed with its axis horizontal on a rough plane inclined at an angle \(\alpha\) to the horizontal. It is held in that position by a light string which is attached at one end to a point on the middle section of the cylinder, passes round part of the circumference and is held at the other end so that it lies entirely in a vertical plane and the free part makes an angle \(\theta\) with the horizontal. The coefficient of friction between the cylinder and the plane is \(\mu\). Show that the tension in the string will be the minimum necessary to maintain equilibrium when \(\theta\) is equal to \(\alpha\) or \(\cos^{-1}\left(\frac{\sin(\alpha-\lambda)}{\sin \lambda}\right)\) according as tan \(\lambda\) is greater or less than \(\frac{1}{\mu}\) tan \(\alpha\).

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A particle of mass \(m\) is suspended from a fixed support by a light elastic string which extends by unit distance under a tension \(\kappa\). Motion of the particle in a vertical line is resisted by a frictional force which is given at any instant by \(cv\), where \(c\) is a constant and \(v\) is the velocity of the particle at that instant. The particle is displaced vertically from the position of equilibrium and released. Show that the subsequent motion will be oscillatory provided that \(c\) is less than \(2\sqrt{m\kappa}\). Show also that if this condition is satisfied, the distances of successive positions of rest from the equilibrium position will be in geometrical progression; and obtain an expression for their common ratio. It may be assumed that the string does not become slack.